Lower Bound on Weights of Large Degree Threshold Functions

  • Vladimir V. Podolskii
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7318)


An integer polynomial p of n variables is called a threshold gate for the Boolean function f of n variables if for all x ∈ {0, 1} n f(x) = 1 if and only if p(x) ≥ 0. The weight of a threshold gate is the sum of its absolute values.

In this paper we study how large weight might be needed if we fix some function and some threshold degree. We prove \(2^{\Omega(2^{2n/5})}\) lower bound on this value. The best previous bound was \(2^{\Omega(2^{n/8})}\) [12].

In addition we present substantially simpler proof of the weaker \(2^{\Omega(2^{n/4})}\) lower bound. This proof is conceptually similar to other proofs of the bounds on weights of nonlinear threshold gates, but avoids a lot of technical details arising in other proofs. We hope that this proof will help to show the ideas behind the construction used to prove these lower bounds.


Boolean Function Minimal Weight Complexity Measure Ordinal Number Circuit Complexity 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Vladimir V. Podolskii
    • 1
  1. 1.Steklov Mathematical InstituteMoscowRussia

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